Many patterns in Nature and industry arise from the system minimizing an appropriate energy. Torn plastic sheets and growing leaves provide striking examples of pattern forming systems which can transition from single wavelength geometries (leaves) to complex fractal-like shapes (lettuce). These fractal-like patterns seem to have many length scales, i.e. the same amount of extra detail can be seen when looking closer (�??statistical self-similarity�?�). It is a mystery how such complex patterns could arise from energy minimization alone. In this talk I will address this puzzle by showing that such patterns naturally arise from the sheet adopting a hyperbolic non-Euclidean geometry. However, there are many different hyperbolic geometries that the growing leaf could select. I will show using techniques from analysis, differential geometry and numerical optimization that the fractal like patterns are indeed the natural minimizers for the system. I will also discuss the implications of our work to developing shape changing soft matter which can be implemented in soft machines.

Complex interactions occur between tumor and host immune system during cancer development and treatment, and a weak systemic immune response can be prognostic of poor patient outcomes. We strive to not only better understand the dynamic behavior of circulating immune cell populations before and during cancer therapy, but also to monitor these dynamic changes to facilitate real-time prediction of patient outcomes and potentially therapy adaptation. I will provide examples of both theoretical (mathematical) and data-driven (epidemiological) approaches to incorporating established systemic immune markers into clinical decision-making. First, coupling models of local tumor-immune dynamics and systemic T cell trafficking allows us to simulate the evolution of tumor and immune cell populations in anatomically distant sites following local therapy, in turn identifying the optimal treatment target for maximum reduction of global tumor burden. Second, improved understanding of how circulating immune markers vary both within and between individual patients can allow more accurate risk stratification at diagnosis, and personalized prediction of patient response to therapy. The importance of multi-disciplinary collaborations in making predictive and prognostic models clinically relevant will be discussed.

The jellyfish has been the subject of numerous mathematical and physical studies ranging from the discovery of reentry phenomenon in electrophysiology to the development of axisymmetric methods for solving fluid-structure interaction problems. In this presentation, we develop and test mathematical models describing the pulsing dynamics and the resulting fluid flow generated by the upside down jellyfish, Cassiopea. The kinematics of contraction and distributions of pulse frequencies were obtained from videos and used as inputs into numerical simulations. Particle image velocimetry was used to obtain spatially and temporally resolved flow fields experimentally. The immersed boundary method was then used to solve the fluid-structure interaction problem and explore how changes in morphology and pulsing dynamics alter the resulting fluid flow. Unlike pelagic (swimming) jellyfish, there is no evidence of the formation of a train of vortex rings. Instead, significant mixing occurs around and directly above the oral arms and secondary mouths. We found good agreement between the numerical simulations and experiments, suggesting that the presence of porous oral arms induce net horizontal flow towards the bell and mixing.

Genetic intratumoral heterogeneity is a natural consequence of imperfect DNA replication. Any two randomly selected cells, whether normal or cancerous, are therefore genetically different. I will discuss the extent of genetic heterogeneity within untreated cancers with particular regard to its clinical relevance. While genomic heterogeneity within primary tumors is associated with relapse, heterogeneity among treatment�??naïve metastases has not been comprehensively assessed. We analyzed sequencing data for 76 untreated metastases from 20 patients and inferred cancer phylogenies for breast, colorectal, endometrial, gastric, lung, melanoma, pancreatic, and prostate cancers. We found that within individual patients a large majority of driver gene mutations are common to all metastases. Further analysis revealed that the driver gene mutations that were not shared by all metastases are unlikely to have functional consequences. A mathematical model of tumor evolution and metastasis formation provides an explanation for the observed driver gene homogeneity. Last, we found that individual metastatic lesions responded concordantly to targeted therapies in 91% of 44 patients. These data indicate that the cells within the primary tumors that gave rise to metastases are genetically homogeneous with respect to functional driver gene mutations and suggest that future efforts to develop combination therapies have the capacity to be curative.

Israel experienced an outbreak of wild poliovirus type 1 (WPV1) in 2013-14, detected through environmental surveillance of the sewage system. No cases of acute flaccid paralysis were reported, and the epidemic subsided after a bivalent oral polio vaccination (bOPV) campaign. As we approach global eradication, polio will increasingly be detected only through environmental surveillance. However, we have lacked the theory to translate environmental surveillance into public health metrics; it is a priori unclear how much environmental surveillance can even say about population-level disease dynamics. We developed a framework to convert quantitative polymerase chain reaction (qPCR) cycle threshold data into scaled WPV1 and OPV1 concentrations for inference within a deterministic, compartmental infectious disease transmission model. We used differential algebra and profile likelihood techniques to perform identifiability analysis, that is, to assess how much information exists in the data for the model, and to quantify inference uncertainty. From the environmental surveillance data, we estimated the epidemic curve and transmission dynamics, determining that the outbreak likely happened much faster than previously thought. Our mathematical modeling approach brings public health relevance to environmental data that, if systematically collected, can guide eradication efforts.

I will discuss some old and new results related to the analysis of stochastic SIR-type epidemics on a configuration model (CM) random graph having a fixed degree distribution p_k. In particular, I will describe the relevant large graph limit result which yields the law of large numbers (LLN) for the edge-based process. I will also discuss the applications of the LLN approximation in building a "network-free" SIR Markov hybrid model which can be used for epidemic parameters inference. The hybrid model idea appears particularly relevant in the context of the recent Ebola and the Zika epidemics.

Graphs are becoming a favorite mathematical object for representation of data. Yet, statistical pattern recognition has focused almost entirely on vector valued data in Euclidean space. Graphs, however, live in graph space, which is non-Euclidean. Thus, most inference techniques are not even defined for graph valued data. Previous work in the classification of graph-valued data typically follows one of two recipes. (1) Vectorize the adjacency matrices of the graphs, and apply standard machine learning techniques. (2) Compute some number of graph invariants (e.g., clustering coefficient, or degree distribution) for each graph, and then apply standard machine learning techniques. We follow a different recipe based in the probabilistic theory of pattern recognition. First, we define a joint graph-class model. Given this model, we derive classifiers which we prove are consistent; that is, they converge to the Bayes optimal classifier. Specifically, we build two consistent classifiers for graph valued data, a parametric and a non-parametric version. In a sense, these classifiers span the spectrum of complexity, the former is consistent for graphs sampled from relatively simple random graph distributions, the latter is consistent for graphs sampled from (nearly) any random graph distribution. Although both classifiers assume that all our graphs have labeled vertices, we generalize these results to also incorporate unlabeled graphs, as well as weighted and multigraphs. We apply these graph classifiers to human brain data. Specifically, using diffusion MRI, we can obtain large brain-graphs (10,000 vertices) for each subject, where vertices correspond to voxels. We then coarsen the graphs spatially to obtain smaller (70 vertex) graphs per subject. Using <50 subjects, we are able to achieve nearly 85% classification accuracy, with results interpretable to neurobiologists with regard to the brain regions of interest.

A conference in honor of Harold Layton's 60th Birthday (Afternoon Session)

This talk will trace many years of work mathematical modeling hematological diseases. The �?understand�? part talks about the use of mathematical to figure out what causes cyclical neutropenia, and the �?treat�? part refers to work on treating cyclical neutropenia using recombinant cytokines. The �?avoid�? part deals with current ongoing work trying to obviate the deleterious effects of chemotherapy on blood cell production�?one of the major negative side effects of chemotherapy.

The evolution of resistance remains an elusive problem in the treatment of both cancer and infectious disease, and represents one of the most important medical problems of our time. While the illnesses are different on several non-trivial levels including timescale and complexity, the underlying biological phenomenon is the same: Darwinian evolution. To comprehensively approach these problems, I have focussed my attention on building a broad suite of investigations centered around the causes and consequences of the evolutionary process in these contexts. I will discuss my and my collaborator's efforts to; model the evolutionary process on the genomic scale in both an analytic (Markov process) and stochastic (individual based model and inference) format; to quantify in vitro competition and interaction between cancer cell lines through an evolutionary game theoretic lens using time-lapse microscopy and computer vision; and to understand the evolutionary contingencies inherent in collateral sensitivity in E. coli and ALK mutated non-small cell lung cancer.

Vaccines protect their recipients by inducing long-term structural changes in populations of immune cells. Part of that restructuring is exactly analogous to Darwinian Selection. New antibody molecules are created by somatic mutation of existing antibody genes. Subsequently, the immune cell populations that possess these mutated receptors overtake the "wild-type" immune cells due to the selective advantage they have acquired. Thus the immune system is vastly better prepared to recognize and eliminate the eliciting pathogen the next time around. New sequencing and biosynthesis technologies, together with mathematical and computational tools, now allow us to investigate this fascinating and important phenomenon more deeply than ever before. I will illustrate this development with examples from the immune response to HIV infection. Second lecture will focus on specifically mathematical questions.

This talk is about extending classical limit theorems of probability (law of large numbers, central limit theorem) to a non-Euclidean setting. I'll talk about new and interesting phenomena observed when sampling independent points from certain singular geometric spaces. The main result is a limit theorem -- the "sticky central limit theorem" -- which applies to the mean or barycenter of a family of independent samples as the number of samples grows. The theorem shows that the geometry of the underlying space may have an interesting effect on the asymptotic fluctuations of the sample means, in a way that does not occur with independent samples in Euclidean space. One motivation for thinking about statistics in singular geometric spaces comes from evolutionary biology; one can consider phylogenetic trees as points in a metric space of the sort discussed in this talk. Apart from this basic motivation, however, the talk will have little biological content and will be mainly about probability.

It is common to use a multitype branching process to model the accumulation of mutations that leads to cancer progression, metastasis, and resistance to treatment. In this talk I will describe results about the time until the first type k (cell with k mutations) and the growth of the type k population obtained in joint work with Stephen Moseley, and their use in evaluating possible screening strategies for ovarian cancer, work in progress with Duke undergraduate Kaveh Danesh. The point process representation of the limit, which is a one-sided stable law, together with results from 10-60 years ago leads to remarkable explicit formulas for Simpson's index and the size of the largest clone. These results are important in understanding tumor diversity which can present serious obstacles to treatment. The last topic is joint work with Jasmine Foo, Kevin Leder, John Mayberry, and Franziska Michor

Ductal carcinoma in situ (DCIS) is an early stage non-invasive breast cancer that originates in the epithelial lining of the milk ducts, but it can evolve into comedo DCIS and ultimately, into the most common type of breast cancer, invasive ductal carcinoma. Understanding the progression and how to effectively intervene in it presents a major scientific challenge. The extracellular matrix surrounding a duct contains several types of cells and several types of growth factors that are known to individually affect tumor growth, but at present the complex biochemical and mechanical interactions of these stromal cells and growth factors with tumor cells is poorly understood. We will discuss a mathematical model that incorporates the cross-talk between stromal and tumor cells, and which can predict how perturbations of the local biochemical and mechanical state influence tumor evolution. We focus on the EGF and TGF-$\beta$ signaling pathways and show how up- or down-regulation of components in these pathways affects cell growth and proliferation, and describe a hybrid model for the interaction of cells with the tumor microenvironment. The analysis sheds light on the interactions between growth factors, mechanical properties of the ECM, and feedback signaling loops between stromal and tumor cells, and suggests how epigenetic changes in transformed cells affect tumor progression.

Cancer emerges due to an evolutionary process in somatic tissue. The fundamental laws of evolution can best be formulated as exact mathematical equations. Therefore, the process of cancer initiation and progression is amenable to mathematical investigation. Of special importance are changes that occur early during malignant transformation because they may result in oncogene addiction and represent promising targets for therapeutic intervention. Here we describe a mathematical approach, called Retracing the Evolutionary Steps in Cancer (RESIC), to deduce the temporal sequence of genetic events during tumorigenesis from crosssectional genomic data of tumors at their fully transformed stage. When applied to a dataset of 70 advanced colorectal cancers, our algorithm accurately predicts the sequence of APC, KRAS, and TP53 mutations previously defined by analyzing tumors at different stages of colon cancer formation. We further validate the method with glioblastoma and leukemia sample data and then apply it to complex integrated genomics databases, finding that high-level EGFR amplification appears to be a late event in primary glioblastomas. RESIC represents the first evolutionary mathematical approach to identify the temporal sequence of mutations driving tumorigenesis and may be useful to guide the validation of candidate genes emerging from cancer genome surveys.

Considering cancer as an evolutionary disease, we aim at understanding the means by which cancer cell populations develop resistance mechanisms to drug therapies, in order to circumvent them by using optimised therapeutic combinations. Rather than focusing on molecular mechanisms such as overexpression of intracellular drug processing enzymes or ABC transporters that are responsible for resistance at the individual cell level, we propose to introduce abstract phenotypes of resistance structuring cancer cell populations. The models we propose rely on continuous adaptive dynamics of cell populations, and are amenable to predict asymptotic evolution of these populations with respect to the phenotypic traits of interest. Drug-induced drug resistance, the question we are tackling from a theoretical and experimental point of view, may be due to biological mechanisms of different natures, mere local regulation, epigenetic modifications (reversible, nevertheless inheritable) or genetic mutations (irreversible), according to the extent to which the genome of the cells in the population is affected. In this respect, the models we develop are more likely to be biologically corresponding to epigenetic modifications, although eventual induction of emergent resistant cell clones due to mutations under drug pressure is not to be completely excluded. From the biologist's point of view, we study phenotypically heterogeneous, but genetically homogeneous, cancer cell populations under stress by drugs. According to the cell populations at stake and to the exerted drug pressure, is drug resistance in cancer a permanently acquired phenotypic trait or is it reversible? Can it be avoided or overcome by rationally (model-guided) designed combinations of drugs? These are some of the questions we will try to answer in a collaboration between a team of mathematicians and another one of biologists, both dealing with cancer and Darwinian - possibly also Lamarckian - evolution of cell populations.

Insulin resistance, a major factor in type 2 diabetes development, is a systemic defect characterized by reduced intracellular insulin signaling. Although there are many proposed causes of insulin resistance, the precise mechanisms that influence its long-term progression remain unclear. In this talk, we develop mathematical models to study the hypothesized role of oxidative stress and mitochondrial dysfunction in skeletal muscle insulin resistance. Simulation results suggest that a perfect storm of environmental and genetic factors leading to oxidative stress can confer protection on the individual cell via insulin resistance.

Viscoelastic interactions of Myxococcus xanthus cells in a low-density domain close to the edge of a swarm have been recently studied in [1] using a combination of a cell-based three-dimensional Subcellular Element (SCE) model [1,2] and cell-tracking experiments. The model takes into account the flexible nature of M. xanthus as well as the effects of adhesion between cells arising from the interaction of the capsular polysaccharide covering two cells in contact with each other. New image and dynamic cell curvature analysis algorithms were used to track and measure the change in cell shapes that occur as flexible cells undergo significant bending during collisions resulting in direct calibration of the model parameters. It will be shown in this talk that flexibility of cells and the adhesive cell��?cell and cell��?substrate interactions of M. xanthus together with cell to aspect-ratio and directional reversals [3], play an important role in smooth cell gliding and more efficient swarming. In the second part of the talk results of the analysis of the three dimensional structures of fibrin networks, with and without cells, reconstructed from two-dimensional z-stacks of confocal microscopy sections using novel image analysis algorithms, will be presented. These images were used to establish microstructure-based models for studying the relationship between the structural features and the mechanical properties of the fibrin networks in blood clots. The change in the fibrin network alignment under applied strain and the elastic modulus values will be shown to agree well with the experimental data [4]. 1. C.W. Harvey, F. Morcos, C.R. Sweet, D. Kaiser, S. Chatterjee, X. Lu, D. Chen and M. Alber [2011], Study of elastic collisions of M. xanthus in swarms, Physical Biology 8, 026016. 2. C.R. Sweet, S. Chatterjee, Z. Xu, K. Bisordi, E.D. Rosen and M. Alber [2011], Modeling Platelet-Blood Flow Interaction Using Subcellular Element Langevin Method, J R Soc Interface, 2011 May 18. [Epub ahead of print], doi: 10.1098/rsif.2011.0180. 3. Y. Wu, Y. Jiang, D. Kaiser and M. Alber [2009], Periodic reversal of direction allows Myxobacteria to swarm, Proc. Natl. Acad. Sci. USA 106 4 1222-1227. 4. E. Kim, O.V. Kim, K.R. Machlus, X. Liu, T. Kupaev, J. Lioi, A.S. Wolberg, D.Z. Chen, E.D. Rosen, Z. Xu and M. Alber [2011], Soft Matter 7, 4983-4992.

Connecting dynamic models with data to yield predictive results often requires a variety of parameter estimation, identifiability, and uncertainty quantification techniques. These approaches can help to determine what is possible to estimate from a given model and data set, and help guide new data collection. Here, we examine how parameter estimation and disease forecasting are affected when examining disease transmission via multiple types or pathways of transmission. Using examples taken from the West Africa Ebola epidemic, HPV, and cholera, we illustrate some of the potential difficulties in estimating the relative contributions of different transmission pathways, and show how alternative data collection may help resolve this unidentifiability. We also illustrate how even in the presence of large uncertainties in the data and model parameters, it may still be possible to successfully forecast disease dynamics.

The ability of animals to self-organize into remarkable patterns of movement is seen throughout nature from herds of large mammals, to flocks of birds, schools of fish, and swarms of insects. Remarkably, patterns of collective movement can be seen even in the simplest forms of life such as bacteria. M. xanthus are common soil bacteria that are among the most �?social" bacteria in nature. In this talk clustering mechanism of swarming M. xanthus will be described using combination of experimental movies and stochastic model simulations. Continuous limits of discrete stochastic dynamical systems simulating cell aggregation will be described in the form of reaction-diffusion and nonlinear diffusion equations. Surface motility such as swarming is thought to precede biofilm formation during infection. Population of bacteria P. aeruginosa, major infection in hospitals, will be shown to efficiently propagate as high density waves that move symmetrically as rings within swarms towards the extending tendrils. Multi-scale model simulations suggest a mechanism of wave propagation as well as branched tendril formation at the edge of the population that depend upon competition between the changing viscosity of the bacterial liquid suspension and the liquid film boundary expansion caused by Marangoni forces. This collective mechanism of cell- cell coordination was recently shown to moderate swarming direction of individual bacteria to avoid a toxic environment. In the last part of the talk a three-dimensional multiscale modeling approach will be described for studying fluid�?viscoelastic cell interaction during blood clot formation.

In this talk, I will discuss a new result in fluid flows through channels with permeable membranes with simple pumping dynamics. Fluid will be exchanged and metabolized in a simple reservoir and I will demonstrate the existence of optimal reservoir properties that may either maximize or minimized the amount of fluid being extracted across the channel walls. The biological relevance of this work may be seen by noting that all living organisms of a sufficient size rely on complex systems of tubular networks to efficiently collect, transport and distribute nutrients or waste. These networks exchange material with the interstitium via embedded channels leading to effective permeabilities across the wall separating the channel interior from the interstitium. In many invertebrates, for example, respiratory systems are made of complex tracheal systems that branch out through the entire body allowing for passive exchange of oxygen and carbon dioxide. In many of these systems, certain animals utilize various pumping mechanisms that alter the flow of the air or fluid being transported. Although the net effect of pumping of the averaged rates of fluid flow through the channel is typically well understood, it is still a largely open problem to understand how, and in what circumstances, pumping enables and enhances the exchange of material across channel walls. It has been demonstrated experimentally, for example, that when certain insects flap their wings, compression of the trachea allow for more efficient oxygen extraction, however it is unclear if this pumping is optimized for flight, oxygen uptake or neither, and understanding this problem quantitatively will shed insight on this biological process. Many of these interesting scenarios occur at low Reynolds number and this regime will be the focus of the presentation.

Carcinogenesis is the transformation of normal cells into cancer cells. This process has been shown to be of a multistage nature, with stem cells that go through a series of (stochastic) genetic and epigenetic changes that eventually lead to a malignancy. Since the origins of the multistage theory in the 1950s, mathematical modeling has played a prominent role in the investigation of the mechanisms of carcinogenesis. In particular, two stochastic (mechanistic) models, the Armitage-Doll and the two-stage clonal expansion (TSCE) model, have been widely used in the past for cancer risk assessment and for the analysis of cancer population and experimental data. In this talk, I will introduce some of the biological and mathematical concepts behind the theory of multistage carcinogenesis, and discuss in detail the use of these models in cancer epidemiology and cancer prevention and control. Recent applications of multistage and state-transition Markov models to assess the potential impact of lung cancer screening in the US will be reviewed.

Circadian rhythms are biological oscillations that align our physiology and behavior with the 24-hour environmental cycles conferred by the Earth�??s rotation. In this talk, I will discuss two projects that focus on circadian clock cells in the brain and the entrainment of circadian rhythms to the light-dark cycle. Most of what we know about the electrical activity of circadian clock neurons comes from studies of nocturnal (night-active) rodents, hindering the translation of this knowledge to diurnal (day-active) humans. In the first part of the talk, we use data assimilation and patch-clamp recordings from the diurnal rodent Rhabdomys pumilio to build the first mathematical models of the electrophysiology of circadian neurons in a day-active species. We find that the electrical activity of circadian neurons is similar overall between nocturnal and diurnal rodents but that there are some interesting differences in their responses to inhibition. In the second part of the talk, we use tools from dynamical systems theory to study the reentrainment of a model of the human circadian pacemaker following perturbations that simulate jet lag. We show that the reentrainment dynamics are organized by invariant manifolds of fixed points of a 24-hour stroboscopic map and use these manifolds to explain a rapid reentrainment phenomenon that occurs under certain jet lag scenarios.

Elastic capsules occur in nature in the form of cells and vesicles and are manufactured for biomedical applications. They are widely modeled but there are few analytical results. In this talk, complex variable techniques are used to derive semi-analytic solutions for the steady-state response and time-dependent evolution of elastic capsules in 2D Stokes flow. The analysis is complemented by spectrally accurate numerical simulations of the time-dependent evolution. One motivation for this work is to provide analytical solutions to help validate the accuracy of numerical methods for elastic membranes in flow. A second motivation is to clarify the steady response of capsules in some canonical flows. Finally, we investigate the formation of finite-time cusp singularities, of which there are only a few examples in interfacial Stokes flow, and where none involve elastic interfaces. This is joint work with Michael Booty and Michael Higley.

Biological oscillators such as the cell cycle, circadian clocks, and metabolic rhythms are ubiquitous across the domains of life. These biochemical oscillators co-exist in the same cells, often sharing and competing for resources. Are there mechanisms and regulatory principles that ensure harmony between these oscillators? Recent studies have shown that in addition to the transcriptional circadian clock, many organisms (including Arabidopsis) have a circadian redox rhythm driven by the organism's metabolic activities. It has been hypothesized that the redox rhythm is linked to the circadian clock, but the mechanism and the biological significance of this link have only begun to be investigated. In the first half of my talk, I will describe our work (in collaboration with the Dong lab at Duke) on the coupling of redox rhythms and the plant circadian clock. In the second half of my talk, I will discuss our work on the coupling of yeast metabolic cycle and the cell division cycle.

Many mathematical models in biology can be described by a system of hyperbolic conservation laws with nonlinear and nonlocal coefficients. In order to determine these coefficients one needs to solve auxiliary systems of equations, for example elliptic or parabolic PDEs, which are coupled to the hyperbolic equations. In this talk we give several examples: The growth of tumors, the shrinking of dermal wounds, T cell differentiation, the growth of drug resistant bacteria in hospitals, and the transport of molecules along microtubules in axon. In these examples, the auxiliary systems range from elliptic-parabolic free boundary problems to nonlocal ODEs. Motivated by biological questions, we shall describe mathematical results regarding properties of the solutions of the conservation laws. For example, we shall determine the stability of spherical tumors and the growth of �?fingers;�? we shall discuss conditions for shrinking of the wound; suggest how to reduce the growth of drug resistant bacteria, and derive biologically motivated asymptotic behavior of solutions.

Population geneticists study the dynamics of alternative genetic types in a replicating population. Most theoretical work rests on a simple Markov chain, called the Wright-Fisher model, to describe how an allele's frequency changes from one generation to the next. We have introduced a broad class of Markov models that share the same mean and variance as the Wright-Fisher model, but may otherwise differ. Even though these models all have the same variance effective population size, they encode a rich diversity of alternative forms of genetic drift, with significant consequences for allele dynamics. We have characterized the behavior of standard population-genetic quantities across this family of generalized models. The generalized population models can produce startling phenomena that differ qualitatively from classical behavior -- such as assured fixation of a new mutant despite the presence of genetic drift. We have derived the forward-time continuum limits of the generalized processes, analogous to Kimura's diffusion limit of the Wright-Fisher process. Finally, we have shown that some of these exotic models are more likely than the Wright-Fisher model itself, given empirical data on genetic variation in Drosophila populations. Joint work with Ricky Der and Charlie Epstein.

Inferential summaries of tree estimates are useful in the setting of evolutionary biology, where phylogenetic trees have been built from DNA data since the 1960's. In bioinformatics, psychometrics and data mining, hierarchical clustering techniques output the same mathematical objects, and practitioners have similar questions about the stability and `generalizability' of these summaries. I will present applications of the Billera, Holmes, Vogtman (2001) distance to inferential problems both in the frequentist (bootstrap) and Bayesian contexts. I will compare the tree of trees representation to the Euclidean approximations of treespace made available through Multidimensional Scaling of the matrix of distances between trees. We also provide applications of the distances between trees to hierarchical clustering trees constructed from microarrays and phylogenetic trees of metagenomic data of bacteria in the gut. This talk contains joint work with John Chakerian and Alfred Spormann.